Depletion interaction between spheres immersed in a solution of ideal polymer chains

JOURNAL OF CHEMICAL PHYSICS VOLUME 113, NUMBER 23 15 DECEMBER 2000

Depletion interaction between spheres immersed in a solutionof ideal polymer chains

Remco Tuiniera)

Van’t Hoff Laboratory, Debye Research Institute, University of Utrecht, Padualaan 8, 3584 CH Utrecht,The Netherlands

Gerrit A. VliegenthartSchool of Chemistry, University of Bristol, Cantock’s Close, Bristol, BS8 1TS United Kingdom

Henk N. W. LekkerkerkerVan’t Hoff Laboratory, Debye Research Institute, University of Utrecht, Padualaan 8, 3584 CH Utrecht,The Netherlands

~Received 28 June 2000; accepted 19 September 2000!

The depletion interaction between two spheres due to nonadsorbing ideal polymers is calculatedfrom the polymer concentration profile using the excess~negative! adsorption. Computersimulations show that the polymer concentration profiles around two spheres are well described bythe product function of the concentration profile around a single sphere. From the interactionpotential between two spheres the second osmotic virial coefficient,B2 , is calculated for variouspolymer-colloid size ratios. We find that when the polymers become smaller than the spheres,B2

remains positive in the dilute regime. This shows that the depletion interaction is ineffective forrelatively small spheres. ©2000 American Institute of Physics.@S0021-9606~00!50247-5#

olae-

x

iedla

ulle

siu

-thhe

e-

foon

lloi-

iesthe-

s,

tebe-meresmerDe

bility

odes ifto

oins

se-

urn aionthe

ouro

tivema

I. INTRODUCTION

The polymer-induced depletion interaction between cloidal particles has been studied extensively during thedecades.1–9 The pioneering work on depletion interaction btween particles due to nonadsorbing polymers was doneAsakura and Oosawa,1 and further elaborated by Vrij,2 andJoannyet al.3 Subsequently, their concepts were verified eperimentally by Vincent and co-workers,4,5 and De Hek andVrij. 6 Since the early eighties, depletion interaction theorof mixtures of spheres and polymer chains were extendethe sense that complete phase diagrams can be calcuwithin a perturbation framework.7,8 The topology of thephase diagrams, which only depends on the ratioRg /R,whereR is the colloid radius andRg is the radius of gyration,is qualitatively described very well by these theories, andsupported by computer simulations.10,11 More recently thetheoretical work has been extended, taking into accopolydispersity.12–14 Most of these theoreticaapproaches2,7,8,12–14replace the polymer chains by penetrabhard spheres~PHS! in order to simplify the polymer chainand take the radius of gyration of the polymers as the radof the PHS.

Milling and Biggs15 found quantitative agreement between the depletion force predicted theoretically andforce measured between a silica plate and a silica spimmersed in a cyclohexane solution of PDMS forR/Rg

'400. They showed that the range of the interaction pottial is very close to 2Rg , which corroborates that the depletion layer thickness is close to the radius of gyrationrelatively very small polymer chains. For experiments

a!Author to whom correspondence should be addressed. [email protected]

10760021-9606/2000/113(23)/10768/8/$17.00

Downloaded 16 Apr 2004 to 131.211.152.81. Redistribution subject to AI

l-st

by

-

sinted

is

nt

s

ere

n-

r

systems with polymers that are much smaller than the codal spheres~the ‘‘colloid limit’’ !, the PHS treatment worksout well.16 However, we also compared these PHS theorto experiments where the colloid radius is of the order ofpolymer radius of gyration,17–20 and found that the theoretical curves lie far below the experimental ones.

If the polymers become much larger than the colloidthe PHS approach is known to fail.21–28 In that limit, alsoreferred to as the ‘‘protein limit,’’ the colloids can penetrathe polymer coil and a proper description of the systemcomes complicated, since the spatial structures of polychains have to be taken into account properly. De Genn21

showed that dispersing small spheres in a semidilute polysolution does not lead to a phase separation. FollowingGennes, a range of studies was done to describe the staof colloid–polymer solutions in the protein limit.22–28 Thesestudies are relevant, for instance, for the stability of foproducts where proteins, which can be regarded as spherglobular, and polysaccharides are often mixed, giving risedepletion-induced phase separations.29–32

In this paper we study the interaction between twspheres immersed in an ideal polymer solution of long chafor arbitrary polymer–colloid size ratios. As a tool we uthe adsorption method,33–35which uses the polymer concentration profiles to calculate the~negative! adsorption, andhence the interaction between colloidal particles. Omethod uses a superposition approximation resulting iproduct function that describes the polymer concentratprofile between two plates or two spheres. We will useinteraction between two plates~Sec. II!, calculated exactlyusing the force method, as a reference case to testmethod. It will be shown that the interaction between twplates calculated using the force method is in quantitail:

8 © 2000 American Institute of Physics

P license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

inem

lasr–rasa

esb

,

a

ldte

Fo

e

aver

raa

is

n,ea,per

ion

thees

e,

leres-ob-

sedbe-

10769J. Chem. Phys., Vol. 113, No. 23, 15 December 2000 Depletion interaction between spheres

agreement with that of a negative adsorption method usthe product of the polymer concentration profiles. Furththe polymer concentration profiles were tested using coputer simulations. In Sec. III, we will apply the~negative!adsorption method to the case of two spheres and calcuthe second osmotic virial coefficient of two spheres afunction of the polymer concentration for various polymecolloid size ratios. Above a certain polymer–colloid sizetio, we find that the depletion interaction is too weak to cauphase separation and thus the colloid–polymer mixturesstable.

II. INTERACTIONS BETWEEN TWO FLAT PLATESIMMERSED IN SOLUTION OF IDEAL POLYMERCHAINS

A. Interaction potential between two flat plates usingthe force method

The force per unit area between two parallel flat platK(h), is the difference between the osmotic pressuretween ~inside! the platesP i , and outside the plates,Po

5kTnb , wherenb is the bulk number density of polymersand reads1

K~h!5nbkTS X~h!211h]X~h!

]h D , ~1!

whereX(h) is defined as

X~h!58

p2 (p51,3,5,...

1

p2 expS 2p2p2Rg

2

h2 D , ~2!

where the radius of gyration of ideal chains is definedlA(N/6), wherel is the segment size, andN is the number oflinks between the segments. Integration of this force yiethe interaction potential per unit area between the plaW(h)

W~h!

kT52nbS X~h!h2h1

4Rg

ApD . ~3!

For penetrable hard spheres, Vrij2 found

WPHS~h!

kT52nb~sPHS2h!, ~4!

wheresPHS is the diameter of a penetrable hard sphere.h50 we find from Eqs.~3! and ~4! that sPHS is equal to4Rg /Ap if we match the minima of the potentials. Both thpredictions of Eqs.~3! and ~4! are plotted in Fig. 1, and theagreement is very good forh,3Rg/2. Forh.2Rg the inter-action potentials deviate strongly; ideal polymer chains ha longer range of attraction than the penetrable hard sphEisenriegler36 noted that Eq.~3! is identical to Eq.~4! up toand including terms of the order ofh4. PHS are thus verygood replacements for ideal chains to describe the intetions for flat walls, and hence are equally expected to behvery well for very large spheres.


gr,-

tea

-ere

,e-

s

ss

r

ees.

c-ve

B. Interaction potential between two plates using theadsorption method

An alternative way to obtain the interaction potentialby making use of the generalized Gibbs equation~see, forinstance, Refs. 33–35, 37!

2S ]W~h!

]m Dh

5G~h!2G~`!, ~5!

wherem is the chemical potential of the polymer solutioandG(h) is the adsorbed amount of polymers per unit ari.e., the number of polymer chains between the platesunit area when the distance between the plates equalsh. Theadsorption is the integral over the polymer concentratprofile n(x)

G~h!5E0

h

dx@n~x!2nb#. ~6!

Since for ideal chainsm5kT ln(nb), and @G(h)2G(`)#;nb , Eq. ~6! yields

W~h!52kT@G~h!2G~`!#. ~7!

Thus, the profile of the polymer concentration betweentwo plates yields the interaction potential. If the two plattouch, the negative adsorption is zero@G(0)50#, andW(0)5kTG(`). From Eq. ~3! we find W(0)52kTnb4Rg /Ap, which says thatG(`)52nb4Rg /Ap522nbj, wherej is the depletion layer thickness per platwhich thus equals 2Rg /Ap. We find from Eqs.~7! and ~3!for G(h)

G~h!5nb~X~h!h2h!. ~8!

In order to calculate the polymer concentration profibetween two flat platesn(x), we first focus on the polymeconcentration profile near a single plate. Analytical exprsions for a polymer in solution near an interface can be

FIG. 1. Normalized interaction potential between two hard plates immerin an ideal polymer solution as a function of the normalized distancetween the platesh/Rg . The predictions for ideal polymer chains@Eq. ~3!# isgiven by the solid curve and that of penetrable hard spheres@PHS; Eq.~4!#is given by the dashed curve.


rohinis

-thon

y

-n

eon

thpr

b

e

te

esutT

gi

-

rdio

sthen

nc-

ra-osi-terac-

bolsrves

using

10770 J. Chem. Phys., Vol. 113, No. 23, 15 December 2000 Tuinier et al.

tained using an analogy between a Brownian diffusion pcess and the path of subsequent segments in a polymer cusing proper boundary conditions. The relation describthe resulting probability function of a polymer chaintermed Edwards’ diffusion equation.38–41 Eisenriegler,42 andlater Marques and Joanny,43 calculated the polymer concentration near one flat plate for ideal chains, and foundfollowing expression for the relative polymer concentratif (x)5n(x)/nb :

f ~x!5n~x!

nb52 erf~z!2erf~2z!1

4z

Ap@exp~2z2!

2exp~24z2!#18z2F1

22erf~2z!1

1

2erf~z!G , ~9!

wherez is defined asx/(2Rg), andx is the distance from thesurface. One can characterize the negative adsorption bdepletion layer thickness,j, defined as

j5E0

`

dxS 12n~x!

nbD . ~10!

For the case of ideal polymer chains near a flat plate@Eq. ~9!#we find a depletion layer thickness,j, equal to 2Rg /Ap,which is in agreement with the analysis above Eq.~8!. Wenote here that the simple formn(x)/nb5tanh2(x/j), derivedby De Gennes39 in the context of depletion of polymer segments in a semidilute polymer solution in a good solvedescribes Eq.~9! within an accuracy of 1%.

In order to be able to describe the interaction betwespheres later on, we propose the following product functi

n~x!

nb5 f ~x! f ~h2x!, ~11!

to make a connection between the individual profile ofnonadsorbing ideal polymers near a single plate and thefile between two plates. In Eq.~11! n(x) is the polymerconcentration between the plates, andf (x) and f (h2x) arethe individual profiles, given by Eq.~9!. The concentrationprofile near a single plate, say plate 1, can be expressedBoltzmann factor as:f (x)5n(x)/nb5exp@2V(x)/kT#, whereV(x) is the free energy giving rise to the profile. For thsecond plate, located at a distanceh, we can then write:f (h2x)5n(h2x)/nb5exp@2V(h2x)/kT#. Subsequently,the product function Eq.~11! follows from the superpositionapproximation:Vtot(x)5V(x)1V(h2x).

In order to test Eq.~11! we compare its results withcomputer simulations, as plotted in Fig. 2. The compusimulations ~symbols! were obtained by placing randomwalks of 100 segments in a box containing two flat platAll configurations are weighted with equal probability, bthose crossing the plates are assigned a zero weight.computer simulation setup is consistent with that of Feiand Napper.44 It follows from the results in Fig. 2 that theproduct function@Eq. ~11!; full curves# gives a good description of the computer simulations forh/Rg.3. For h/Rg

,p, the sum ofX(h) of Eq. ~2! is dominated byp51.Then, one can use a ground-state approximation to Edwadiffusion equation to calculate the polymer concentrat


-ain,g

e

the

t,

n:

eo-

y a

r

.

hen

s’n

profile for small gap widths. For that situation the largeeigenvalue dominates in the expansion and the profile is tdescribed as39–41

n~x!

nb5

16

p2 expS 2p2Rg

2

h2 D sin2S px

h D . ~12!

So, for small gap widths we can compare the product fution with the results of Eq.~12!. In Fig. 2 we have plottedboth the predictions of Eq.~11! ~solid curves! as well asthose of Eq.~12! for h/Rg52, and 1.5~dashed curves!. Ob-viously, Eq. ~11! overestimates the ground-state Eq.~12!.Thus, the relative deviation is large for small plate sepations, which agrees with the expectation that the superption approximation could be deficient for very small plaseparations. However, the absolute deviation for the inte

FIG. 3. Normalized adsorbed amountG(h) as a function of the normalizedplate separationh/Rg , using the exact result@solid curve; Eq.~8!#, and fromthe product function for the profile@Eqs.~6!, ~9!, and~11!#.

FIG. 2. Polymer concentration profiles between two flat plates. The symcorrespond to the computer simulations for 100 segments. The full cuare the predictions according to the product function Eq.~11! @using Eq.~9!#, and the dashed curves are the ground-state approximation resultsEq. ~12!.


m

in

hete

eauuf

tes.wseretwo

ion

rns


tion is small; the distance between the plates becomes sand thereforeG(h)/G(`)→0 for smallh @see Eq.~6!#.

In order to test whether the product function@Eq. ~11!#yields a good prediction of the adsorptionG(h), we com-pared it with the exact result of Eq.~8! in Fig. 3. The exactresult is plotted as the solid curve and the adsorption, usthe product function as calculated from inserting Eq.~11!into Eq. ~6! is plotted as the dashed curve in Fig. 3. Tresults are generally consistent but differ somewhat for inmediate plate separations. We conclude here that Eq.~11! isa function that describes the main features of the polymconcentration profile, and hence the resulting adsorptioninteraction between two flat plates. It follows that the calclation of the depletion layer thickness for a single plate s

tee

e


all

g

r-

rnd--

fices to describe the depletion interaction between two plaWe therefore assume that the depletion layer that follofrom the polymer concentration profile near a single sphcan also be used to describe the interactions betweenspheres. First, we will focus on the polymer concentratprofile near a single sphere.

III. SPHERES IMMERSED IN AN IDEAL POLYMERCHAIN SOLUTION

A. Ideal polymer chains near a single sphere

Taniguchi et al.,45 and independently Eisenriegleet al.,22 found the concentration of ideal polymer chaiaround a single hard sphere with radiusR which reads

f ~x!5S R

R12zRgD 23

S 2zRg

R D 2

1S zRg

R D S erf~z!22z2~12erf~z!!12

Apz exp~2z2!D

12 erf~z!2erf~2z!14z

Ap@exp~2z2!2exp~24z2!#

18z2F1

22erf~2z!1

1

2erf~z!G 4 , ~13!

ly-

he

tamleree-

by a

where z again equalsx/2Rg , and wherex is the distancefrom the surface of the sphere. We first note that Eq.~13!approaches Eq.~9! in the limit R@Rg . In the limit of verysmall particle radii, we arrive at the limitn(x)/nb5(x/$R1x%)25(12R/$R1x%)2. A similar limit was also found byOdijk25 for a small colloidal sphere immersed in a semidilupolymer solution with a correlation length being much larg

FIG. 4. Polymer depletion layer profiles for ideal nonadsorbing polymchains near a flat plate@solid curve; Eq.~9!# and near a sphere@dashedcurves; Eq.~13!# as a function of the distance from the surfacex. The resultsof the computer simulations are given for the flat plate~open diamonds!, andthe single sphere forR/Rg50.1 ~crosses!, 1 ~open circles!, and 10~closedsquares!.

r

than the colloid radius. In the ‘‘protein’’ limit whereRg

@R, the profile becomes independent of the relevant pomer length scale.

For various ratios ofRg /R we plotted the profiles as afunction of the distance from the interface in Fig. 4 as tdashed curves, as well as the single plate result~solid curve!of Eq. ~9!. In Fig. 4 we also added computer simulation da~symbols!. The simulations were obtained by placing randowalks in a box containing a single flat plate or a singsphere. We observe that the simulation results are in agment with Eqs.~9! and~13!. For large radii (Rg /R50.1) we

r

FIG. 5. Schematic picture of the geometry of two spheres separateddistanceh. The relevant parameters are indicated in the figure.


e

fs

wepoto

imereoo

yof

e

(ra-

ainslts

fair

res,l

spheres immersed in a polymer solution using the product


see that the sphere profile approaches that of a flat plate@Eq.~9!#. For Rg /R51 the depletion layer thickness already bcomes significantly smaller than 2Rg /Ap. For very smallspheres (Rg /R510), the depletion layer is of the order othe sphere radius, rather than of the order of the radiugyration. ForRg@R we find a limiting value ofj'3R.

B. Interaction between two spheres immersed in asolution with ideal polymers

In Sec. II we showed that the product function@Eq. ~11!#describes the interactions between two flat plates quitein comparison with exact expressions for the interactiontential of two plates. Here, we apply the product functioncalculate the interactions between spherical colloidsmersed in a polymer solution. First, we calculate the polymconcentration profile in the space surrounding two spheimmersed in an ideal polymer solution. A schematic twdimensional picture of the situation is given in Fig. 5 for twspheres with a center-to-center distance ofh12R. We as-sume that the local polymer concentrationns(r ) in everypoint defined by the vectorr is given by a product function


-

of

ll-

-rs

-

ns~r !

nb5 f s~r 1! f s~r 2!, ~14!

where f s(r i) is the polymer concentration profile given bEq. ~13!, and r i is the closest distance to the surfacespherei. Several computer simulation results~symbols! forR/Rg510 are plotted in Figs. 6~a!–6~d! as a function ofz~normalized byRg) and are compared with the result of thproduct function given by Eq.~14!. In Fig. 6~a! we focus onthe profiles along sphere 1 while sphere 2 touches ith50). For h/Rg50.77, 3.1, and 7 we present the concenttion profiles between the spheres in Figs. 6~b!–~d! for vari-ous values ofx ~normalized byRg!. For the situation wherethe spheres have a comparable size as the polymer ch(R/Rg51.4), we also present computer simulation resu~symbols! for h/Rg50.77, and 3.1 in Figs. 7~a! and ~b!, re-spectively. In general, we observe that the profiles are inagreement with the predictions of the product function.

In order to arrive at the interactions between the sphewe calculate the~negative! adsorption around two colloida

he
FIG. 6. ~a! Polymer concentration profiles in the region around two spheres forR/Rg510, h50 andx57, 10, and 13. The symbols are the results from tcomputer simulations. The curves correspond to the result of the product function expressed by Eq.~14!. ~b! As ~a! for h/Rg50.77, andx50.5, 5, 7, and 10.~c! As ~a! for h/Rg53.1. ~d! As ~a! for h/Rg57.

he

nghiin

he

lsap

q.

ofn

n,

Inforac-in

thensta-ffi-of

et

deal

-


function to describe the profiles around the particles. Tnegative adsorption is now the excess number of polymper volume and is denoted asN(h), whereh is the closestsphere-to-sphere distance. From the~negative! adsorption wecan calculate the interaction potential from Eq.~8!. It followsthat the interaction potential between two spheres,Ws(h), isrelated toN(h) as

Ws~h!

kT5N~`!2N~h!. ~15!

The result forR/Rg5100 is plotted in Fig. 8~dashed curve!.We normalized the interaction potential curves by dividiover the absolute value of the minimum of the curves. In tlarge sphere limit we can apply the Derjaguapproximation46 to the result of the force method@Eq. ~4!#

Ws~h!5Eh

`

dh8 pRW~h8!, ~16!

FIG. 7. ~a! Polymer concentration profiles forR/Rg51.4 in the regionaround two spheres forh/Rg50.77 andx51, 2, 3, and 3.5. The symbols arthe results from the computer simulations. The curves correspond toresult of the product function expressed by Eq.~14!. ~b! As ~a! for h/Rg

53.1.


ers

s

which is plotted as the solid curve in Fig. 8. The depth of tminimum of the potential (h50) was derived byEisenriegler36 and reads

Ws~0!

nbkT524pRRg

2 ln~2! orWs~0!

kT523fpS R

RgD ln~2!.

~17!

Here,fp is the relative polymer concentration which equa4pnbRg

3/3 and is defined such that it is unity at the overlconcentration. From our numerical calculations using E~15! we find Ws(0)/kT52227fp for R/Rg5100, whichcan be compared withWs(0)/kT52208fp , as followsfrom Eq. ~17!. We can thus conclude that the minimumthe interaction potential from our numerical integratioagrees with the Derjaguin approximation to Eq.~4! within10%. In comparison with the result of the product functiowe observe~see Fig. 8! that the range of Eq.~16! is some-what smaller than our product function approximation.Fig. 9 we plotted the results from the product functionsR/Rg510, 3, 1, and 0.3. We observe that the range of attrtion becomes smaller with decreasing colloid radius,agreement with a decreasing depletion layer thickness.

C. Consequences for the colloidal stability

The final aim is to understand the consequences ofdepletion interaction for the stability of colloidal suspensiocontaining nonadsorbing polymer chains. The colloidal sbility can be expressed by the second osmotic virial coecient, which is a measure of the effective excluded volumethe colloids. The second osmotic virial coefficient,B2 , isdefined as

B252pE0

`

r 2@12exp~2W~r !/kT!#dr, ~18!

he

FIG. 8. Interaction potential between two spheres immersed in an ipolymer solution normalized by the depth of the minimum forR/Rg5100.The dashed curve is the result from the adsorption method@Eq. ~15!# usingthe product function@Eqs. ~14!#. The solid curve is the result of the Derjaguin approximation to the result of the force method@combining Eqs.~4!and ~16!#.


co

be

stiv

en

easoi

eto

Othe

p-ery

theon-

othenc-iththelsores.

with

d

ita

n ofion


wherer is now the center-to-center distance between theloidal spheres. For hard spheresB2 is four times the volumeof a particle, and~depletion-induced! attraction will reduceB2 ~at sufficiently high polymer concentrations!. When B2

becomes sufficiently negative a colloidal suspensioncomes unstable.47

In Fig. 10 we plottedB2 as a function of the relativepolymer concentration for four colloid–polymer size ratioWe see that the depletion interaction becomes ineffecwhen the polymers become larger than the colloids;B2 be-comes sufficiently negative only when the polymer conctration is close to the overlap concentration. ForR50.3Rg

the interaction is ineffective to such a degree thatB2 is al-ways positive in the dilute regime. This explains why rathhigh polymer concentrations are required to induce phseparation in systems with a relatively large polymer–collsize ratio.17–20

FIG. 9. Interaction potentials as in Fig. 8 between two spheres immersean ideal polymer solution forR/Rg510, 3, 1, and 0.3.

FIG. 10. Second osmotic virial coefficient of hard spheres, normalized wthe sphere volumeVs , immersed in a solution of ideal polymer chains asfunction of the relative polymer concentration forR/Rg510, 3, 1, and 0.3.


l-

-

.e

-

re

d

It is illustrative to compare our results with thAsakura–Oosawa~AO! potential between two spheres duedepletion of penetrable hard spheres, as derived by Vrij.2 Wehave shown thatsPHS is equal to 4Rg /Ap @see below Eq.~4!#. Consequently, we find for the AO~PHS! potential fortwo spheres

WsAO~r !52fpS R

Rg1

2

ApD 3

3F 123

4 S r

Rg

R

Rg1

2

Ap

D 11

16S r

Rg

R

Rg1

2

Ap

D 3G .

~19!

Insertion of this potential into Eq.~18! yields the resultsplotted in Fig. 11. It is clear that forR/Rg510 and 3 theresults for the second osmotic virial coefficient with the Apotential are quite comparable with those obtained withproduct function. However, forR/Rg51, and especially forR/Rg50.3, the deviation is quite significant. The PHS aproach predicts that phase separation would occur at vlow polymer concentrations for spheres smaller thanpolymer coils. This shows that a PHS approach is a reasable approximation only as long asR/Rg.1.

IV. CONCLUSIONS

Calculations of the interaction potential between twplates immersed in an ideal polymer solution showed thatinteractions could be described in terms of the product fution of the single plates only. The results are consistent wcomputer simulations. Therefore, we proposed thatdepletion layer profile around an individual sphere could abe used to describe the interactions between two spheThe approximated profiles near two spheres also agreecomputer simulations. A calculation of the~negative! ad-

in

h

FIG. 11. Second osmotic virial coefficient of hard spheres as a functiothe relative polymer concentration as in Fig. 10 for the AO interactpotential.


lubaclaceenti

ist,eey

hy

d P

m.

ki,

ns. 1

ys.


sorption around two spheres in a very dilute polymer sotion containing ideal chains is shown to be in reasonaagreement with the Derjaguin approximation of the exresult. With increasing polymer–colloid size ratio, the retive range of the interaction decreases. The polymer contration dependence of the second osmotic virial coefficishows that the depletion interaction becomes less effecfor R.Rg with increasing polymer–colloid size ratio.

ACKNOWLEDGMENTS

The NWO-Unilever program financially supported thwork. We would like to thank W. J. Briels, J. K. G. DhonG. J. Vroege, P. van der Schoot, and R. Wensink for thhelp and useful discussions. We thank Dr. T. Taniguchi, Dpartment of Computational Science & Engineering, NagoUniversity, Japan, for supplying his M.Sc. thesis.

1S. Asakura and F. Oosawa, J. Chem. Phys.22, 1255~1954!.2A. Vrij, Pure Appl. Chem.48, 471 ~1976!.3J. F. Joanny, L. Leibler, and P. G. De Gennes, J. Polym. Sci. Pol. P17, 1073~1979!.

4O. Li-In, B. Vincent, and F. A. Waite, ACS Symp. Ser.9, 165 ~1975!.5B. Vincent, P. F. Luckham, and F. A. Waite, J. Colloid Interface Sci.73,508 ~1980!.

6H. De Hek, and A. Vrij, J. Colloid Interface Sci.84, 409 ~1981!.7A. P. Gast, C. K. Hall, and W. B. Russel, J. Colloid Interface Sci.96, 251~1983!.

8H. N. W. Lekkerkerker, W. C. K. Poon, P. N. Pusey, A. Stroobants, anB. Warren, Europhys. Lett.20, 559 ~1992!.

9W. C. K. Poon, Curr. Opin. Colloid Interface Sci.3, 593 ~1998!.10E. J. Meijer and D. Frenkel, J. Chem. Phys.100, 6873~1994!.11M. Dijkstra, J. M. Brader, and R. Evans, J. Phys.: Condens. Matter11,

10079~1999!.12X. L. Chu, A. D. Nikolov, and D. T. Wasan, Langmuir12, 5004~1996!.13P. B. Warren, Langmuir13, 4588~1997!.14M. Piech and J. Y. Walz, J. Colloid Interface Sci.225, 134 ~2000!.15A. Milling and S. Biggs, J. Colloid Interface Sci.170, 604 ~1995!.16S. M. Ilett, A. Orrock, W. C. K. Poon, and P. N. Pusey, Phys. Rev. E51,

1344 ~1995!.


-let

-n-t

ve

ir-a

s.

.

17M. A. Faers and P. F. Luckham, Langmuir15, 2922~1997!.18E. H. A. De Hoog and H. N. W. Lekkerkerker, J. Phys. Chem. B103,

5274 ~1999!.19I. Bodnar, J. K. G. Dhont, and H. N. W. Lekkerkerker, J. Phys. Che

100, 19614~1996!.20I. Bodnar and W. D. Oosterbaan, J. Chem. Phys.106, 7777~1997!.21P. G. De Gennes, C.R. Acad. Sci. B288, 359 ~1979!.22E. Eisenriegler, A. Hanke, and S. Dietrich, Phys. Rev. E54, 1134~1996!.23A. Hanke, E. Eisenriegler, and S. Dietrich, Phys. Rev. E59, 6853~1999!.24E. Eisenriegler, J. Phys.: Condens. Matter12, A227 ~2000!.25T. Odijk, Macromolecules29, 1842~1996!.26T. Odijk, Physica A278, 347 ~2000!.27P. van der Schoot, Macromolecules31, 4635~1998!.28A. M. Kulkarni, A. P. Chatterjee, K. S. Schweizer, and C. F. Zukos

Phys. Rev. Lett.83, 4554~1999!.29V. Ya. Grinberg, and V. B. Tolstoguzov, Food Hydrocolloids11, 145

~1997!.30A. Syrbe, W. J. Bauer, and H. Klostermeyer, Int. Dairy J.8, 179 ~1998!.31R. Tuinier and C. G. de Kruif, J. Chem. Phys.110, 9296~1999!.32R. Tuinier, J. K. G. Dhont, and C. G. de Kruif, Langmuir16, 1497~2000!.33D. G. Hall, J. Chem. Soc., Faraday Trans. 168, 2169~1972!.34S. G. Ash, D. H. Everett, and C. Radke, J. Chem. Soc., Faraday Tra

69, 1256~1973!.35R. Evans, B. Marini, and U. Marconi, J. Chem. Phys.86, 7138~1987!.36E. Eisenriegler, Phys. Rev. E55, 3116~1997!.37Y. Mao, P. Bladon, H. N. W. Lekkerkerker, and M. E. Cates, Mol. Ph

92, 151 ~1997!.38S. F. Edwards and K. F. Freed, J. Phys. A2, 145 ~1969!.39P. G. De Gennes,Scaling Concepts in Polymer Physics~Cornell Univer-

sity Press, Ithaca, 1979!.40M. Doi and S. F. Edwards,The Theory of Polymer Dynamics~Clarendon

Oxford, 1986!.41A. Y. Grosberg and A. R. Khoklov,Statistical Mechanics of Macromol-

ecules~AIP, New York, 1994!.42E. Eisenriegler, J. Chem. Phys.79, 1052~1983!.43C. M. Marques and J. F. Joanny, Macromolecules23, 268 ~1990!.44R. I. Feigin and D. H. Napper, J. Colloid Interface Sci.71, 117 ~1979!.45T. Taniguchi, T. Kawakatsu, and K. Kawasaki,Slow Dynamics in Con-

densed MatterAIP Series, edited by K. Kawasaki,256, 503 ~1992!.46B. V. Deryaguin, Kolloid-Z.69, 155 ~1934!.47G. A. Vliegenthart and H. N. W. Lekkerkerker, J. Chem. Phys.112, 5364

~2000!.


https://www.researchgate.net/publication/237109900_Scaling_Concept_In_Polymer_Physics?el=1_x_8&enrichId=rgreq-81b802711ba72aa36d12b5684c9086ef-XXX&enrichSource=Y292ZXJQYWdlOzQ2NjQyOTY1O0FTOjEwMTA3ODMyMjM4NDkxOUAxNDAxMTEwMzU4OTE4

https://www.researchgate.net/publication/237109900_Scaling_Concept_In_Polymer_Physics?el=1_x_8&enrichId=rgreq-81b802711ba72aa36d12b5684c9086ef-XXX&enrichSource=Y292ZXJQYWdlOzQ2NjQyOTY1O0FTOjEwMTA3ODMyMjM4NDkxOUAxNDAxMTEwMzU4OTE4

Depletion interaction between spheres immersed in a solution of ideal polymer chains

Documents

Transcript of Depletion interaction between spheres immersed in a solution of ideal polymer chains